Persistence and Permanence of Mass-Action and Power-Law Dynamical Systems

Abstract

Persistence and permanence are properties of dynamical systems that describe the long-term behavior of the solutions, and in particular specify whether positive solutions approach the boundary of the positive orthant. Mass-action systems (or more generally power-law systems) are very common in chemistry, biology, and engineering, and are often used to describe the dynamics in interaction networks. We prove that two-species mass-action systems derived from weakly reversible networks are both persistent and permanent, for any values of the reaction rate parameters. Moreover, we prove that a larger class of networks, called endotactic networks, also give rise to permanent systems, even if we allow the reaction rate parameters to vary in time. These results also apply to power-law systems and other nonlinear dynamical systems. In addition, ideas behind these results allow us to prove the Global Attractor Conjecture for three-species systems. ).1 See Definition 2.5. 2 The version in [9] only required that no positive trajectory converges to a boundary point; we conjecture that even more is true: no positive trajectory has an ω-limit point on the boundary.

A Geometric Approach to the Global Attractor Conjecture

“…In order to simplify the computations in section 6, we differ from the usual convention [15,39] by letting init w (Y ) denote the ≤ w -maximal elements rather than the ≤ w -minimal elements. Accordingly, the inequalities in Definition 3.14 are switched, so our definition of endotactic is equivalent to the usual one.…”

“…See Figure 2. The dual picture to these ideas was explained in [15,Proposition 4.1] by way of the so-called parallel sweep test.…”

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

“…The dynamical theory of persistence has been extensively used in biological population dynamics, ecology, epidemiology, chemical reactions, game theory, neural networks and other important areas of applied sciences and engineering. The references Anderson [2], Bonneuil [3], Butler and Wolkowicz [4], Calzada et al [5], Cantrell and Cosner [6], Craciun et al [7], Hale and Waltman [19], Hofbauer and Sigmund [22], Johnston et al [25], Smith and Thieme [49] and Takeuchi [50] illustrate some of the classical and also more recent applications of the mentioned theory in these fields.…”

Coexistence in molecular communications

“…To verify that the equilibrium is complex balanced, we use the deficiency zero theorem (detailed in Appendix C). The fact that this reaction network converges to a unique equilibrium follows from the global attractor theorem for three-species networks [26].…”